φ-FEM: a finite element method on domains defined by level-sets
Michel Duprez
∗and Alexei Lozinski
†March 27, 2020
Abstract
We propose a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we search the approximation to the solution as a product of a finite element function with the given level-set function, which is also approximated by finite elements. Unlike other recent fictitious domain-type methods (XFEM, CutFEM), our approach does not need any non-standard numerical integration (on cut mesh elements or on the actual boundary). We consider the Poisson equation discretized with piecewise polynomial Lagrange finite elements of any order and prove the optimal convergence of our method in the H1-norm. Moreover, the discrete problem is proven to
be well conditioned, i.e. the condition number of the associated finite element matrix is of the same order as that of a standard finite element method on a comparable conforming mesh. Numerical results confirm the optimal convergence in both H1and L2norms.
1
Introduction
We consider the Poisson-Dirichlet problem
−∆u = f on Ω, u = 0 on Γ, (1)
in a bounded domain Ω ⊂ Rd (d = 2, 3) with smooth boundary Γ assuming that Ω and Γ are given by a
level-set function φ:
Ω := {φ < 0} and Γ := {φ = 0}. (2)
Such a representation is a popular and useful tool to deal with problems with evolving surfaces or interfaces [17]. In the present article, the level-set function is supposed to be known on Rd, smooth, and to behave near Γ as the signed distance to Γ. We propose a finite element method for the problem above which is easy to implement, does not require a mesh fitted to Γ, and is guaranteed to converge optimally. Our basic idea is very simple: one cannot impose the Dirichlet boundary conditions in the usual manner since the boundary Γ is not resolved by the mesh, but one can search for the approximation to u as a product of a finite element function wh with the level-set φ itself: such a product obviously vanishes on Γ. In order to
make this idea work, some stabilization should be added to the scheme as outlined below and explained in detail in the next section. We coin our method φ-FEM in accordance with the tradition of denoting the level-sets by φ.
More specifically, let us assume that Ω lies inside a simply shaped domain O (typically a box in Rd)
and introduce a quasi-uniform simplicial mesh TO
h on O (the background mesh). Let Th be a submesh of
TO
h obtained by getting rid of mesh elements lying entirely outside Ω (the definition of Th will be slightly
∗CEREMADE, Universit´e Paris-Dauphine & CNRS UMR 7534, Universit´e PSL, 75016 Paris, France.
mduprez@math.cnrs.fr
†Laboratoire de Math´ematiques de Besan¸con, UMR CNRS 6623, Universit´e Bourgogne Franche-Comt´e, 16, route de Gray,
changed afterwords). Denote by Ωh the domain covered by the mesh Th (so that typically Ωh is only
slightly larger than Ω). Our starting point is the following formal observation: assuming that the right-hand side f is actually well defined on Ωh, and the solution u can be extended to Ωh so that −∆u = f on
Ωh, we can introduce the new unknown w ∈ H1(Ωh) such that u = φw and the boundary condition on Γ
is automatically satisfied. An integration by parts yields then Z Ωh ∇(φw) · ∇(φv) − Z ∂Ωh ∂ ∂n(φw)φv = Z Ωh f φv, ∀v ∈ H1(Ω h). (3)
Given a finite element approximation φhto φ on the mesh Thand a finite element space Vhon Th, one can
then try to search for wh ∈ Vh such that the equality in (3) with the subscripts h everywhere is satisfied
for all the test functions vh ∈ Vh and to reconstruct an approximate solution uh to (1) as φhwh. These
considerations are very formal and, not surprisingly, such a method does not work as is. We shall show however that it becomes a valid scheme once a proper stabilization in the vein of the Ghost penalty [3] is added. The details on the stabilization and on the resulting finite element scheme are given in the next section.
Our method shares many features with other finite elements methods on non-matching meshes, such as XFEM [15, 14, 18, 11] or CutFEM [5, 6, 7, 4]. Unlike the present work, the integrals over Ω are kept in XFEM or CutFEM discretizations, which is cumbersome in practice since one needs to implement the integration on the boundary Γ and on parts of mesh elements cut by the boundary. The first attempt to alleviate this practical difficulty was done in [12] with method that does not require to perform the integration on the cut elements, but needs still the integration on Γ. In the present article, we fully avoid any non trivial numerical integration: all the integration in φ-FEM is performed on the whole mesh elements, and there are no integrals on Γ. We also note that an easily implementable version of φ-FEM is here developed for Pk finite elements of any order k ≥ 1. This should be contrasted with the situation
in CutFEM where some additional terms should be added in order to achieve the optimal Pk accuracy if
k > 1, cf. [8]. An alternative approach avoiding non trivial quadrature is presented in a recent work on the shifted boundary method [13]. The optimal convergence with piecewise linear finite elements (k = 1) on a non-fitted mesh is achieved there by introducing a truncated Taylor expansion on the approximate boundary.
The article is structured as follows: our φ-FEM method is presented in the next section. We also give there the assumptions on the level-set φ and on the mesh, and announce our main result: the a priori error estimate for φ-FEM. We work with standard continuous Pk finite elements on a simplicial mesh and prove
the optimal order hk for the error in the H1norm and the (slightly) suboptimal order hk+1/2for the error
in the L2 norm.1 The proofs of these estimates are the subject of Section 3. We contend ourselves with
the error analysis pertinent to the h-refinement only, i.e. we do not attempt to track the dependence of the constants, appearing in our estimates, on the polynomial degree. In Section 4, we prove that the linear system produced by our method has the condition number of order 1/h2, the same as that of a standard
finite element method. Numerical illustrations are given in Section 5, including a test case covered by our theory that confirms the theoretical predictions and other test cases going slightly beyond the theoretical framework. Finally, conclusions and perspectives are presented in Section 6.
2
Definitions, assumptions, description of
φ-FEM, and the main
result
We recall that we work with a bounded domain Ω ⊂ O ⊂ Rd (d = 2, 3) with boundary Γ given by a
level-set φ as in (2). We assume that φ is sufficiently smooth and behaves near Γ as the signed distance to Γ after an appropriate change of local coordinates. More specifically, we fix an integer k ≥ 1 and introduce the following
1Our approach can also be realized using Qkfinite elements on a mesh consisting of rectangles/cubes. The convergence
Assumption 1. The boundary Γ can be covered by open sets Oi, i = 1, . . . , I and one can introduce
on every Oi local coordinates ξ1, . . . , ξd with ξd = φ such that all the partial derivatives ∂αξ/∂xα and
∂αx/∂ξα up to order k + 1 are bounded by some C
0> 0. Moreover, φ is of class Ck+1 on O and |φ| ≥ m
on O \ ∪i=1,...,IOi with some m > 0.
Let TO
h be a quasi-uniform simplicial mesh on O of mesh size h, meaning that hT ≤ h and ρ(T ) ≥ βh
for all simplexes T ∈ TO
h with some mesh regularity parameter β > 0 (here hT = diam(T ) and ρ(T ) is the
radius of the largest ball inscribed in T ). Consider, for an integer l ≥ 1, the finite element space Vh,O(l) = {vh∈ H1(O) : vh|T ∈ Pl(T ) ∀T ∈ ThO}
where Pl(T ) stands for the space of polynomials in d variables of degree ≤ l viewed as functions on T .
Introduce an approximate level-set φh∈ Vh,O(l) by
φh:= Ih,O(l) (φ) (4)
where Ih,O(l) is the standard Lagrange interpolation operator on Vh,O(l). We shall use this to approximate the physical domain Ω = {φ < 0} with smooth boundary Γ = {φ = 0} by the domain {φh< 0} with the
piecewise polynomial boundary Γh= {φh= 0}. We employ φh rather than φ in our numerical method in
order to simplify its implementation (all the integrals in the forthcoming finite element formulation will involve only the piecewise polynomials). This feature will also turn out to be crucial in our theoretical analysis.
We now introduce the computational mesh Thas the subset of ThOcomposed of the triangles/tetrahedrons
having a non-empty intersection with the approximate domain {φh< 0}. We denote the domain occupied
by Th by Ωh, i.e.
Th:= {T ∈ ThO : T ∩ {φh< 0} 6= ∅} and Ωh= (∪T ∈ThT )
o.
Note that we do not necessarily have Ω ⊂ Ωh. Indeed some mesh elements can be cut by the exact
boundary {φ = 0} but not by the approximate one {φh = 0}. In such rare occasions, a mesh element
containing a small portion of Ω will not be included into Th.
Fix an integer k ≥ 1 (the same k as in Assumption 1) and consider the finite element space Vh(k)= {vh∈ H1(Ωh) : vh|T ∈ Pk(T ) ∀ T ∈ Th}.
The φ-FEM approximation to (1) is introduced as follows: find wh∈ Vh(k) such that:
ah(wh, vh) = lh(vh) for all vh∈ Vh(k), (5)
where the bilinear form ah and the linear form lh are defined by
ah(w, v) := Z Ωh ∇(φhw) · ∇(φhv) − Z ∂Ωh ∂ ∂n(φhw)φhv + Gh(w, v) (6) and lh(v) := Z Ωh f φhv + Grhsh (v),
with Gh and Grhsh standing for
Gh(w, v) := σh X E∈FΓ h Z E ∂ ∂n(φhw) ∂ ∂n(φhv) + σh2 X T ∈TΓ h Z T ∆(φhw)∆(φhv) , Grhsh (v) := −σh2 X T ∈TΓ h Z T f ∆(φhv) ,
where σ > 0 is an h-independent stabilization parameter, TΓ
h ⊂ Th contains the mesh elements cut by the
approximate boundary Γh= {φh= 0}, i.e.
ThΓ= {T ∈ Th: T ∩ Γh6= ∅}, ΩΓh:= ∪T ∈TΓ hT o . (7) and FΓ
h collects the interior facets of the mesh Theither cut by Γh or belonging to a cut mesh element
FhΓ= {E (an internal facet of Th) such that ∃T ∈ Th: T ∩ Γh6= ∅ and E ∈ ∂T }.
The brackets inside the integral over E ∈ FΓ
h in the formula for Gh stand for the jump over the facet E.
The first part in Gh actually coincides with the ghost penalty as introduced in [3] for P1 finite elements.
We shall also need the following assumptions on the mesh Th, more specifically on the intersection of
elements of Th with the approximate boundary Γh= {φh= 0}.
Assumption 2. The approximate boundary Γh can be covered by element patches {Πi}i=1,...,NΠ having the following properties:
• Each patch Πi is a connected set composed of a mesh element Ti∈ Th\ ThΓ and some mesh elements
cut by Γh. More precisely, Πi= Ti∪ ΠΓi with ΠΓi ⊂ ThΓ containing at most M mesh elements;
• TΓ h = ∪
NΠ
i=1ΠΓi;
• Πi and Πj are disjoint if i 6= j.
Assumption 2 is satisfied for h small enough, preventing strong oscillations of Γ on the length scale h. It can be reformulated by saying that each cut element T ∈ TΓ
h can be connected to an uncut element
T′ ∈ T
h\ ThΓ by a path consisting of a small number of mesh elements adjacent to one another; see [12]
for a more detailed discussion and an illustration (Fig. 2).
In what follows, k · kk,D (resp. | · |k,D) denote the norm (resp. the semi-norm) in the Sobolev space
Hk(D) with an integer k ≥ 0 where D can be a domain in Rd or a (d − 1)-dimensional manifold.
Theorem 2.1. Suppose that Assumptions 1 and 2 hold true, l ≥ k, the mesh Th is quasi-uniform, and
f ∈ Hk(Ω
h∪ Ω). Let u ∈ Hk+2(Ω) be the solution to (1) and wh∈ Vh(k) be the solution to (5). Denoting
uh:= φhwh, it holds
|u − uh|1,Ω∩Ωh ≤ Ch
k
kfkk,Ω∪Ωh (8)
with a constant C > 0 depending on the C0, m, M in Assumptions 1, 2, on the maximum of the derivatives
of φ of order up to k + 1, on the mesh regularity, and on the polynomial degrees k and l, but independent of h, f , and u.
Moreover, supposing Ω ⊂ Ωh
ku − uhk0,Ω≤ Chk+1/2kfkk,Ωh (9)
with a constant C > 0 of the same type.
3
Proof of the a priori error estimate
The proof of Theorem 2.1 is preceded with auxiliary lemmas in Sections 3.1 and 3.2, followed by the proof of coercivity of the form ah in Section 3.3.
3.1
A Hardy-type inequality
Lemma 3.1. We assume that the domain Ω is given by the level-set φ, cf. (2), and satisfies Assumption 1. Then, for any u ∈ Hk+1(O) vanishing on Γ,
u φ k,O ≤ Ckukk+1,O
with C > 0 depending only on the constants in Assumption 1. Proof. The proof is decomposed into three steps:
Step 1. We start in the one dimensional setting and adapt the proof oF Hardy’s inequality from [16]. Let u : R → R be a C∞ function with compact support such that u(0) = 0. Set w(x) = u(x)/x for x 6= 0
and w(0) = u′(0). We shall prove that w is a C∞function on R and, for any integer s ≥ 0, Z ∞ −∞|w (s) (x)|2dx 1/2 ≤ C Z ∞ −∞|u (s+1) (x)|2dx 1/2 (10) with C depending only on s.
Observe, for any x > 0,
w(x) = u(x) x = 1 x Z x 0 u′(t)dt = Z 1 0 u′(xt)dt. Hence, w(s)(x) = Z 1 0 u(s+1)(xt)tsdt. (11)
It implies limx→0+w(s)(x) = u(s+1)(0)/(s + 1). The same formula holds for the limit as x → 0−. This means that w is continuous (the special case s = 0), and w(s)(0) exists for all s ≥ 1.
We have now by (11) and the integral version of Minkowski’s inequality Z ∞ 0 |w (s) (x)|2dx 1/2 = Z ∞ 0 Z 1 0 u(s+1)(xt)tsdt 2 dx !1/2 ≤ Z 1 0 Z ∞ 0 |u (s+1) (xt)|2dx 1/2 tsdt = C Z ∞ 0 |u (s+1) (x)|2dx 1/2
with C =R01ts−1/2dt = 1/(s + 1/2). Applying the same argument to negative x, we get (10).
Step 2. Let now u : Rd → R be a compactly supported C∞ function vanishing at x
d = 0 and set
w = u/xd. We shall prove
|w|k,Rd≤ C|u|k+1,Rd (12)
with C depending only on k.
To keep things simple, we give here the proof for the case d = 2 only (the case d = 3 is similar but would involve more complicated notations). Take any integers t, s ≥ 0 with t + s = k, apply (10) to ∂tw
∂xt 1 = 1 x2 ∂tu ∂xt 1 treated as a function of x2 (note that ∂
tu
∂xt
1 vanishes at x2= 0) and then integrate with respect to x1. This gives ∂kw ∂xt 1∂xs2 0,Rd ≤ C ∂k+1u ∂xt 1∂x s+1 2 0,Rd . Thus, |w|2k,Rd= k X s=0 ∂kw ∂xk−s1 ∂xs2 2 0,Rd ≤ C2 k X s=0 ∂k+1u ∂xk−s1 ∂xs+12 2 0,Rd ≤ C2|u|2k+1,Rd
so that (12) is proved.
Step 3. Consider finally the domains Ω ⊂ O as announced in the statement of this Lemma, let u be a C∞function on O vanishing on Γ, and set w = u/φ. Assume first that u is compactly supported in O
l, one
of the sets forming the cover of Γ as announced in Assumption 1. Recall the local coordinated ξ1, . . . , ξd
on Olwith ξd = φ and denote by ˆu (resp. ˆw) the function u (resp. w) treated as a function of ξ1, . . . , ξd.
Since ˆw = ˆu/ξd, (12) implies k ˆwkk,Rd≤ Ckˆukk+1,Rd. Passing from the coordinates x1, . . . , xd to ξ1, . . . , ξd
and backwards we conclude kwkk,Ol ≤ Ckukk+1,Ol with a constant C that depends on the maximum of partial derivatives ∂αx/∂ξα up to order k and that of ∂αξ/∂xαup to order k + 1. Introducing a partition
of unity subject to the cover {Ol} we can now easily prove kwkk,O≤ Ckukk+1,O noting that 1/φ is of class
Ck outside ∪
l{Ol}. This estimate holds also true for u ∈ Hk+1(O) by density of C∞ in Hk+1.
3.2
Some technical lemmas
This Section regroups some technical results to be used later in the proofs of the coercivity of ah (Section
3.3) and the a priori error estimates (Sections 3.4 and 3.5). The most important contribution here is Lemma 3.3 which extends to finite elements of any degree a result from [12]. This lemma will be the keystone of the proof of coercivity by allowing us to handle the non necessarily positive terms on the cut elements. It shows indeed that the H1 norm of a finite element function on ΩΓh can be bounded by its
norm on the whole computational domain Ωh multiplied by a number strictly smaller than 1, modulo the
addition of stabilization terms. We recall that our stabilization is strongly inspired by that of [3] but differs from it by some extra terms involving the Laplacian on mesh elements. The proof that such a stabilization is sufficient in Lemma 3.3 relies on a simple observation on polynomials, announced and proven in Lemma 3.2.
Lemma 3.2. Let T be a triangle/tetrahedron, E one of its sides and p a polynomial on T of degree s ≥ 0 such that p = ∂p∂n = 0 on E and ∆p = 0 on T . Then p = 0 on T .
Proof. Let us consider only the 2D case (3D is similar). Without loss of generality, we can assume that E lies on the x-axis in (x, y) coordinates. Let p = P pijxiyj with i, j ≥ 0, i + j ≤ s as above. We
shall prove by induction on m = 0, 1, . . . , l that pim = 0, ∀i. Indeed, this is valid for m = 0, 1 since
p(x, 0) =Pipi0xi= 0 and ∂y∂p(x, 0) =Pipi1xi= 0. Now, ∆p = 0 implies for all indices i, j ≥ 0
(i + 2)(i + 1)pi+2,j+ (j + 2)(j + 1)pi,j+2 = 0
so that pim= 0, ∀i implies pi,m+2= 0, ∀i.
Lemma 3.3. Under Assumption 2, for any β > 0 and s ∈ N∗ one can choose 0 < α < 1 depending only on the mesh regularity and s such that, for each vh∈ Vh(s),
|vh|21,ΩΓ h≤ α|vh| 2 1,Ωh+ βh X E∈FΓ h ∂vh ∂n 2 0,E + βh2 X T ∈TΓ h k∆vhk20,T. (13)
Proof. Choose any β > 0, consider the decomposition of ΩΓ
h in element patches {Πk} as in Assumption 2,
and introduce α := max Πk,vh6=0 |vh|21,ΠΓ k− βh P E∈Fk ∂vh ∂n 2 0,E− βh 2P T ⊂Πkk∆vhk 2 0,T |vh|21,Πk , (14)
where the maximum is taken over all the possible configurations of a patch Πk allowed by the mesh
regularity and over all the piecewise polynomial functions on Πk (polynomials of degree ≤ s). The subset
Fk⊂ FhΓ gathers the edges internal to Πk. Note that the quantity under the max sign in (14) is invariant
patch Πk contains at most M elements. Thus, the maximum is indeed attained since it is taken over a
bounded set in a finite dimensional space.
Clearly, α ≤ 1. Supposing α = 1 would lead to a contradiction. Indeed, if α = 1 then we can take Πk,
vh yielding this maximum and suppose without loss of generality |vh|1,Πk= 1. We observe then |vh|21,Tk+ βh X E∈Fk ∂vh ∂n 2 0,E + βh2 X T ⊂Πk k∆vhk20,T = 0 since |vh|21,Πk = |vh| 2 1,Tk+ |vh| 2 1,ΠΓ
k. This implies vh = c = const on Tk, ∂vh
∂n
= 0 on all E ∈ Fk, and
∆vh = 0 on all T ⊂ Πk. Thus applying Lemma 3.2 to vh− c, we deduce that vh = c on Πk, which
contradicts |vh|1,Πk= 1.
This proves α < 1. We have thus |vh|21,ΠΓ k≤ α|vh| 2 1,Πk+ βh X E∈Fk ∂vh ∂n 2 0,E + βh2 X T ⊂Πk k∆vhk20,T
for all vh∈ Vhand all the admissible patches Πk. Summing this over Πk, k = 1, . . . , NΠ yields (13).
Lemma 3.4. For all vh∈ Vh(k), it holds
kφhvhk0,ΩΓ
h ≤ Ch |φhvh|1,Ω Γ
h , (15)
kφhvhk0,Ωh\Ω≤ Ch |φhvh|1,Ωh , (16)
with a constant C > 0 depending only on the regularity of Th and k.
Proof. It is easy to see that the supremum
C = sup
ph6=0,T
kphk0,T
hT|ph|1,T
over all the polynomials ph ∈ Pk+l(T ) vanishing at a point of T and all the simplexes T satisfying the
regularity assumption hT/ρ(T ) ≥ β is attained so that C is finite. Taking any T ∈ ThΓ and putting
ph = φhvh, this implies kφhvhk0,T ≤ ChT|φhvh|1,T for any Vh ∈ Vh(k). Summing over all T ∈ ThΓ
concludes the proof of (15). Estimate (16) is proven similarly, adding, if necessary, neighbour elements to T ∈ TΓ
h.
Lemma 3.5. For all vh∈ Vh(k)
X E∈FΓ h kφhvhk20,E ≤ Ch|φhvh|21,Ωh (17) and kφhvhk20,∂Ωh ≤ Ch|φhvh| 2 1,Ωh (18)
with a constant C > 0 depending only on the regularity of Th.
Proof. Let E ∈ FΓ
h. Recall the well-known trace inequality
kvk20,E ≤ C 1 hkvk 2 0,T+ h|v|21,T (19) for each v ∈ H1(E). Summing this over all E ∈ FΓ
h gives X E∈FΓ h kφhvhk20,E≤ C 1 hkφhvhk 2 0,ΩΓ h+ h|φhvh| 2 1,ΩΓ h
Lemma 3.6. Under Assumption 1, it holds for all v ∈ Hs(Ωh) with integer 1 ≤ s ≤ k + 1, v vanishing
on Ω,
kvk0,Ωh\Ω≤ Ch
s
kvks,Ωh\Ω. (20)
Proof. Consider the 2D case (d = 2). For simplicity, we can assume that v is C∞ regular and pass to
v ∈ Hs(Ω
h) by density. By Assumption 1, we can pass to the local coordinates ξ1, ξ2 on every set Ok
covering Γ assuming that ξ1 varies between 0 and L and, for any ξ1 fixed, ξ2 varies on Ωh\ Ω from 0 to
some b(ξ1) with 0 ≤ b(ξ1) ≤ Ch. We observe using the bounds on the mapping (x1, x2) 7→ (ξ1, ξ2)
kvk20,(Ωh\Ω)∩Ok≤ C Z L 0 Z b(ξ1) 0 v2(ξ1, ξ2)dξ2dξ1 (recall that ∂ αv ∂ξα 2 (ξ1, 0) = 0 for α = 0, . . . , s-1 and b ≤ Ch) = C Z L 0 Z b(ξ1) 0 Z ξ2 0 (ξ2− t)s−1 (s − 1)! ∂sv ∂ξs 2 (ξ1, t)dt !2 dξ2dξ1 ≤ C Z L 0 h2s Z b(ξ1) 0 ∂sv ∂ξs 2 (ξ1, t) 2 dtdξ1 ≤ Ch2s|v|2s,(Ωh\Ω)∩Ok.
Summing over all neighbourhoods Ok gives (20). The proof in the 3D case is the same up to the change
of notations.
3.3
Coercivity of the bilinear form
a
hLemma 3.7. Under Assumption 2, the bilinear form ah is coercive on Vh(k) with respect to the norm
|||vh|||h:=
q
|φhvh|21,Ωh+ Gh(vh, vh)
i.e. ah(vh, vh) ≥ c|||vh|||2h for all vh ∈ Vh(k) with c > 0 depending only on the mesh regularity and on the
constants in Assumption 2.
Proof. Let vh∈ Vh(k) and Bh be the strip between Γhand ∂Ωh, i.e. Bh= {φh> 0} ∩ Ωh. Since φhvh= 0
on Γh, Z ∂Ωh ∂(φhvh) ∂n φhvh= Z ∂Bh ∂(φhvh) ∂n φhvh = X T ∈TΓ h Z ∂(Bh∩T ) ∂(φhvh) ∂n φhvh− X T ∈TΓ h X E∈Fcut h (T ) Z Bh∩E ∂(φhvh) ∂n φhvh,
where ThΓ is defined in (7) and Fhcut(T ) regroups the facets of a mesh element T cut by Γh. By divergence
theorem, Z ∂Ωh ∂(φhvh) ∂n φhvh= Z Bh |∇(φhvh)|2+ X T ∈TΓ h Z Bh∩T ∆(φhvh)φhvh − X E∈FΓ h Z E∩Bh φhvh ∂(φhvh) ∂n .
Substituting this into the definition of ah yields ah(vh, vh) = Z Ωh |∇(φhvh)|2− Z Bh |∇(φhvh)|2− X T ∈TΓ h Z Bh∩T ∆(φhvh)φhvh + X F ∈FΓ h Z F ∩Bh φhvh ∂(φhvh) ∂n + σh2 X T ∈TΓ h Z T|∆(φv h)|2+ σh X E∈FΓ h Z E ∂(φhvh) ∂n 2 . (21)
Since Bh⊂ ΩΓh (cf. (7)), applying Lemma 3.3 to φhvh∈ Vh(k+l) gives
Z Bh |∇(φhvh)|2≤ α|φhvh|21,Ωh+ βh X E∈FΓ h Z E ∂(φhvh) ∂n 2 + βh2 X T ∈TΓ h Z T|∆(φ hvh)|2.
Moreover, by Young inequality, (15) and (17), we obtain for any ε > 0 X T ∈TΓ h Z Bh∩T ∆(φhvh)φhvh≤ h2 2ε X T ∈TΓ h Z T|∆(φ hvh)|2+ Cε|φhvh|21,Ωh and X F ∈FΓ h Z F ∩Bh φhvh ∂(φhvh) ∂n ≤2εh X E∈FΓ h Z E ∂(φhvh) ∂n 2 + Cε|φhvh|21,Ωh. Thus, putting the last 3 bounds into (21) we arrive at
a(vh, vh) ≥ (1 − α − Cε) |φhvh|21,Ωh + σ − β − 1 2ε h X E∈FΓ h ∂(φhvh) ∂n 2 0,E + σ − β − 1 2ε h2 X T ∈TΓ h Z T|∆(φv h)|2.
This leads to the conclusion taking ε sufficiently small and σ sufficiently big.
3.4
Proof of the
H
1error estimate in Theorem 2.1
Since the bilinear form ah is coercive, it remains to construct a good interpolant of the exact solution u in
the form of a product of a function from Vh(k)with φh. The details of such a construction are given below
together with the appropriate interpolation estimates. An additional difficulty will come from the extra terms in the Galerkin orthogonality relation (24) with ˜f resulting from the extension of u from Ω to Ωh.
These terms turn out to be of optimal order since ˜f differs from f only on a narrow strip of width ∼ h, cf. Lemma 3.6 and (26).
We now proceed with the detailed proof. Since f ∈ Hk(Ω), the solution u of (1) belongs to Hk+2(Ω)
(see [10, p. 323]) and can be extended by a function ˜u in Hk+2(O), cf. [10, p. 257], such that ˜u = u on
Ω and
k˜ukk+2,Ωh ≤ k˜ukk+2,O≤ Ckukk+2,Ω≤ Ckfkk,Ω. (22) Let w = ˜u/φ. By Lemma 3.1,
|w|k+1,Ωh ≤ Ckukk+2,O≤ Ckfkk,Ω. (23) Introduce the bilinear form ¯ah, similar to ah as defined in (6) but with φ instead of φh multiplying the
trial function: ¯ ah(w, v) = Z Ωh ∇(φw) · ∇(φhv) − Z ∂Ωh ∂ ∂n(φw)φhv + σh X E∈FΓ h Z ∂ ∂n(φw) ∂ ∂n(φhv) + σh2 X T ∈Th Z T ∆(φw)∆(φhv).
Since φw = ˜u ∈ H2(Ω
h), an integration by parts yields
¯ ah(w, vh) = Z Ωh ˜ f φhvh− σh2 X T ∈TΓ h Z T ˜ f ∆(φhvh), ∀vh∈ Vh
with ˜f = −∆˜u on Ωh. Hence,
ah(wh, vh) − ¯ah(w, vh) = Z Ωh (f − ˜f )φhvh− σh2 X T ∈TΓ h Z T(f − ˜ f )∆(φhvh). (24) Put vh= wh− Ihw and rh= φw − φhIhw
with the nodal interpolator Ih. Eq. (24) can be rewritten as
ah(vh, vh) = ¯ah(w, vh) − ah(Ihw, vh) + Z Ωh (f − ˜f )φhvh− σh2 X T ∈TΓ h Z T(f − ˜ f )∆(φhvh) = Z Ωh ∇rh· ∇(φhvh) − Z ∂Ωh ∂rh ∂nφhvh + σh X E∈FΓ h Z ∂rh ∂n ∂ ∂n(φhvh) + σh2 X T ∈TΓ h Z T ∆rh∆(φhvh) + Z Ωh (f − ˜f )φhvh− σh2 X T ∈Th Z T(f − ˜ f )∆(φhvh).
By Lemma 3.7, Young inequality, and recalling f = ˜f on Ω, we now get c|||vh|||2h≤ 1 2ε|rh| 2 1,Ωh+ h 2ε ∂rh ∂n 2 0,∂Ωh +σ 2h 2ε X E∈FΓ h ∂rh ∂n 2 0,E +σ 2h2 2ε X T ∈TΓ h k∆rhk20,T+ (1 + σ2)h2 2ε kf − ˜f k 2 0,Ωh\Ω +ε 2 |φhvh|21,Ωh+ 1 hkφhvhk 2 0,∂Ωh+ h X E∈FΓ h ∂ ∂n(φhvh) 2 0,E +2h2 X T ∈TΓ h k∆(φhvh)k20,T+ 1 h2kφhvhk 2 0,Ωh\Ω .
The terms above multiplied by ε/2 can be absorbed by the left-hand side. Indeed, the first contribution |φhvh|21,Ωh and the sums over F
Γ
h and ThΓ are evidently controlled by |||vh|||2h. The remaining terms are
controlled by |φhvh|21,Ωh and hence by |||vh|||
2
h thanks to (16) and (18). Taking ε small enough, we thus
conclude |||vh|||2h≤ C |rh|21,Ωh+ h ∂ ∂n(φw − φhIhw) 2 0,∂Ωh +h2 X T ∈TΓ h k∆rhk20,T+ h X E∈FΓ h ∂rh ∂n 2 0,E + h2kf − ˜f k20,Ωh\Ω . (25)
We now estimate each term in the right-hand side of (25). By triangular inequality, |rh|1,Ωh≤ |(φ − φh)w|1,Ωh+ |φh(w − Ihw)|1,Ωh ≤ k∇(φ − φh)kL∞ (Ωh)kwk0,Ωh+ kφ − φhkL∞(Ωh)|w|1,Ωh + k∇φhkL∞(Ω h)kw − Ihwk0,Ωh+ kφhkL∞(Ω h)|w − Ihw|1,Ωh. We continue using the classical interpolation bounds (see for instance [2])
|rh|1,Ωh≤ Ch k (|φ|Wk+1,∞(Ωh)kwk0,Ωh+ |φ|Wk,∞(Ωh)|w|1,Ωh + |φ|W1,∞(Ω h)|w|k,Ωh+ kφkL∞(Ωh)|w|k+1,Ωh) ≤ ChkkφkWk+1,∞(Ωh)kwkk+1,Ωh. Similarly, X T ∈Th |rh|22,T !1 2 ≤ Chk−1kφk Wk+1,∞(Ω h)kwkk+1,Ωh. and ∂rh ∂n 2 0,∂Ωh + X E∈FΓ h ∂rh ∂n 2 0,E ≤ Ch2k−1kφk2 Wk+1,∞(Ωh)kwk2k+1,Ωh. Finally, we get by Lemma 3.6 applied to f − ˜f which vanishes on Ω,
kf − ˜f k0,Ωh\Ω≤ Ch k−1 kf − ˜f kk−1,Ωh\Ω≤ Ch k−1 (kfkk−1,Ωh+ k˜ukk+1,Ωh) (26) since ˜f = −∆˜u.
Using all the bounds above in (25), we get |φh(wh− Ihw)|1,Ωh ≤ |||vh|||h≤ Ch
k
(kwkk+1,Ωh+ kfkk−1,Ωh+ k˜ukk+1,Ωh) (27) with a constant C > 0 that has absorbed kφkWk+1,∞(Ω
h). Applying again the the triangle inequality and the interpolation bounds,
|u − φhwh|1,Ω∩Ωh ≤ |˜u − φhwh|1,Ωh
≤ |(φ − φh)w|1,Ωh+ |φh(w − Ihw)|1,Ωh+ |φh(Ihw − wh)|1,Ωh
≤ Chk(kwkk+1,Ωh+ kfkk−1,Ωh+ k˜ukk+1,Ωh). We have thus proven (8) taking into account the bounds (22) and (23).
3.5
Proof of the
L
2error estimate in Theorem 2.1
As usual, the L2 error estimate will be proven here by Aubin-Nitsche trick. However, the discrepancy
between Ω and Ωh, as well as between φ and φh, gives rise to numerous terms, which should be bounded
through rather tedious calculations. We shall skip some repetitive technical details as they are similar to those in the proof of the H1 error estimate above. We also recall that we do not track explicitly the
dependence of constants on the norms of φ. Let z ∈ H3(Ω) be solution to
−∆z = u − uh on Ω, z = 0 on Γ.
Extend it to Ωh by ˜z ∈ H3(Ωh) using an extension operator bounded in the H3norm. Set y = ˜z/φ. Then
thanks to Lemma 3.1 and to the elliptic regularity estimate. We also have By Lemma 3.1 from [12], we have for any v ∈ H1(ΩΓ
h) kvk0,ΩΓ h≤ C √ hkvk0,Γ+ h|v|1,ΩΓ h . (29) Putting v = ∇˜z, (29) gives |˜z|1,ΩΓ h ≤ C √ hk∇˜zk0,Γ+ h|˜z|2,ΩΓ h ≤ C√hk˜zk2,Ωh ≤ C √ hku − uhk0,Ω. (30) By integration by parts, ku − uhk20,Ω= Z Ω(u − u h)(−∆z) = − Z Γ(u − u h) ∂z ∂n+ Z Ω∇(u − u h) · ∇z. (31)
To treat the first term in (31), we remark first − Z Γ(u − u h)∂z ∂n ≤ ku − uhk0,Γ ∂z ∂n 0,Γ ≤ Cku − uhk0,Γku − uhk0,Ω.
Furthermore, since the distance between Γ and Γh is of order hk+1, we have
ku − uhk0,Γ≤ C(k˜u − uhk0,Γh+ h (k+1)/2 |˜u − uh|1,Ωh) = C(k(φ − φh)wk0,Γh+ h (k+1)/2|˜u − u h|1,Ωh) ≤ C(hk+1kwk0,Γh+ h (k+1)/2+k kfkk,Ωh).
We have used here the already proven bound on |˜u − uh|1,Ωh and the interpolation error bound for φ − φh. We have thus thanks to Lemma 3.1,
ku − uhk0,Γ≤ Chk+1(kwk1,Ωh+ kfkk,Ωh) ≤ Chk+1(k˜uk2,Ωh+ kfkk,Ωh) ≤ Ch k+1 kfkk,Ωh. Hence, − Z Γ(u − u h) ∂z ∂n ≤ Ch k+1 kfkk,Ωhku − uhk0,Ω. (32)
The second term in (31) is treated by Galerkin orthogonality (24): for any yh∈ Vh(k)
Z Ω∇(u − u h) · ∇z = Z Ωh ∇(φw − φhwh) · ∇(φy − φhyh) | {z } I − Z Ωh\Ω ∇(φw − φhwh) · ∇(φy) | {z } II + Z ∂Ωh ∂ ∂n(φw − φhwh)(φhyh) | {z } III − σh X E∈FΓ h Z E ∂ ∂n(φw − φhwh) ∂ ∂n(φhyh) − σh2 X T ∈TΓ h Z T∆(φw − φ hwh)∆(φhyh) | {z } IV + Z Ωh (f − ˜f )φhyh− σh2 X T ∈TΓ h Z T(f − ˜ f )∆(φhyh) | {z } V . (33)
We now estimate term by term the right-hand side of the above inequality taking yh = ˜Ihy with ˜Ih the
Cl´ement interpolation operator on Th.
Term I: by Cauchy-Schwartz, the already proven bound on |˜u − uh|1,Ωh, and (28), |I| ≤ C|˜u − uh|1,Ωh|φy − φhyh|1,Ωh ≤ Ch
k+1
kfkk,Ωhkyk2,Ωh
≤ Chk+1kfkk,Ωhku − uhk1,Ω. Term II: using (30) for ˜z = φy,
|II| ≤ |˜u − uh|1,Ωh|˜z|1,Ωh\Ω≤ Ch
k+1/2kfk
k,Ωhku − uhk0,Ω.
Term III: applying the trace inequality (19) on the mesh elements adjacent to ∂Ωhyields
|III| ≤ C X T ∈TΓ h 1 h|˜u − uh| 2 1,T+ |˜u − uh|22,T 1/2 kφhyhk0,∂Ωh.
The term in parentheses can be further bounded using the triangle inequality, interpolation estimates, and the bound (27) on vh= φh(wh− Ihw) as (· · · )1/2≤ 1h|˜u − φhIhw|21,ΩΓ h+ h|˜u − φhIhw| 2 2,T 1/2 +√1 h|||vh|||h ≤ Chk−1/2kfkk,Ωh. Moreover, since ΩΓ
h is a strip around Γh of width ∼ h, we have
kφhkL∞(ΩΓ h)≤ Chk∇φhkL ∞(∂Ω h)≤ Ch (34) and, by (28), kφhyhk0,∂Ωh ≤ Chkyk1,Ωh ≤ Chku − uhk0,Ω so that |III| ≤ Chk+1/2kfkk,Ωhku − uhk0,Ω.
Term IV: by Cauchy-Schwarz and trace inequalities, together with the interpolation estimates, |IV | ≤ (Chkkfkk,Ωh+ |||vh|||h)Gh(yh, yh) 1/2 ≤ Chkkfkk,ΩhGh(yh, yh) 1/2 and by (30) and (34), Gh(yh, yh)1/2 ≤ C hkφhyhk0,ΩΓh ≤ Ckyk0,Ω Γ h ≤ C|˜z|1,Ω Γ h ≤ C √ hku − uhk0,Ω. (35) Hence, |IV | ≤ Chk+1/2kfk k,Ωhku − uhk0,Ω. Term V: by an inverse inequality and (26),
|V | ≤ Ckf − ˜f k0,Ωh\Ωkφhyhk0,Ωh\Ω≤ Ch
k−1
kfkk,Ωhkφhyhk0,Ωh\Ω. Proceeding as in (35) we conclude
Combining the bounds for the terms I–V in (33) with (32) and putting all this into (31), we obtain by Young inequality ku − uhk20,Ω≤ Chk+1kfkk,Ωhku − uhk1,Ω+ Ch k+1/2 kfkk,Ωhku − uhk0,Ω ≤ Cεh2k+1kfk2k,Ωh+ εhku − uhk 2 1,Ω+ εku − uhk20,Ω.
By the already established estimate for |u − uh|1,Ω,
ku − uhk20,Ω≤ C 1 ε+ ε h2k+1kfk2k,Ωh+ (ε + εh)ku − uhk 2 0,Ω
which proves (9) taking sufficiently small ε.
4
Conditioning of the system matrix
We are now going to prove that the condition number of the finite element matrix associated to the bilinear form ahof φ-FEM does not suffer from the introduction of the multiplication by φh: it is of order 1/h2on
a quasi-uniform mesh of step h, similar to the standard FEM on a fitted mesh.
Theorem 4.1(Conditioning). Under Assumptions 1 and 2 and recalling that the mesh Th is supposed to
be quasi-uniform, the condition number defined by κ(A) := kAk2kA−1k2 of the matrix A associated to the
bilinear form ah on Vh(k), as in (6), satisfies
κ(A) ≤ Ch−2.
Here, k · k2 stands for the matrix norm associated to the vector 2-norm | · |2.
Proof. Step 1 (a lower bound on ah). We shall prove for all wh∈ Vh(k)
ah(wh, wh) ≥ Ckwhk20,Ωh. (36)
By Lemma 3.7, it holds for each wh∈ Vh(k)
ah(wh, wh) ≥ c|||wh|||h2 ≥ c|φhwh|21,Ωh.
We now denote uh= φhwhand apply Lemma 3.1 with k = 0 and φhinstead of φ to wh= uh/φh:
kwhk0,Ωh ≤ Ckuhk1,Ωh. (37)
This is justified by a possible relaxation of the hypotheses of Lemma 3.1. The constant in (37) depends on kφhkW1,∞(Ωh) which is bounded uniformly in h. Moreover, the local coordinates around Γ evoked in Assumption 1 can be reused to build the same around Γh.
Applying Poincar´e inequality on the domain Ωin
h := {φh< 0} yields, as uh= 0 on Γh= ∂Ωinh,
kuhk0,Ωin
h ≤ C|uh|1,Ωinh with a constant that depends only on the diameter of Ωin
h and can be thus assumed h-independent.
Moreover, invoking Lemma 3.4 and observing Ωh\ Ωinh ⊂ ΩΓh we conclude that
kuhk0,Ωh ≤ C|uh|1,Ωh. Combining this with (37) we finish the proof of (36) as follows:
ah(wh, wh) ≥ c|uh|21,Ωh ≥ Ckuhk
2
1,Ωh ≥ Ckwhk
2 0,Ωh.
Step 2(an upper bound on ah). We shall prove for all wh∈ Vh(k)
ah(wh, wh) ≤ C
h2kwhk 2
0,Ωh. (38)
By definition of ah and Lemma 3.5,
ah(wh, wh) ≤ C|φhwh|21,Ωh+ C √ h ∂(φhwh) ∂n 0,∂Ωh |φhwh|1,Ωh+ Ch 2 X T ∈TΓ h |φhwh|22,T.
Using the inverse inequalities on Vh(k+l) ∂(φhwh) ∂n 0,∂Ω h ≤√C hkφhwhk0,Ωh, |φhwh|1,Ωh ≤ C hkφhwhk0,Ωh, and |φhwh|2,T ≤hC2kφhwhk0,T yields ah(wh, wh) ≤ Ckφhwhk20,Ωh, which entails (38) since φh is bounded uniformly in h.
Step 3. Denote the dimension of Vh(k)by N and let us associate any vh∈ Vh(k)with the vector v ∈ RN
contaning the expansion coefficients of vh in the standard finite element basis. Recalling that the mesh is
quasi-uniform and using the equivalence of norms on the reference element, we can easily prove that C1hd/2|v|2≤ kvhk0,Ωh ≤ C2h
d/2
|v|2. (39)
The bounds (39) and (38) imply kAk2= sup v∈RN (Av, v) |v|2 2 = sup v∈RN a(vh, vh) |v|2 2 ≤ Ch d sup vh∈Vh a(vh, vh) kvhk20 ≤ Ch d−2.
Similarly, (39) and (36) imply kA−1k2= sup v∈RN |v|2 2 (Av, v) = supv∈RN |v|2 2 a(vh, vh)≤ C hd sup vh∈Vh kvhk20 a(vh, vh) ≤ C hd.
These estimates lead to the desired result.
5
Numerical results
We have implemented φ-FEM in FEniCS Project [1] and report here some results using uniform Cartesian meshes on a rectangle O as the backgound mesh TO
h . The level-sets φ are approximated in all the tests
by the same finite elements as wh, i.e. we take l = k in (4). All the integrals in ah are computed exactly
by using quadrature rules of sufficiently high order.
1st test case
Let Ω be the circle of radius√2/4 centered at the point (0.5, 0.5) with φ(x, y) = 1/8−(x−1/2)2−(y−1/2)2
and the surrounding domain O = (0, 1)2. We use φ-FEM to solve numerically Poisson-Dirichlet problem (1) with the exact solution given by u(x, y) = φ(x, y) × exp(x) × sin(2πy). The errors in L2and H1norms
for φ-FEM with P1 finite elements are reported in Fig. 1. Taking the stabilization parameter σ = 20, the
order in the L2 norm is 2 and is thus better than in theory). We also observe that the ghost stabilization
is crucial to ensure the convergence of the method, cf. the results with σ = 0. The condition numbers are reported in Fig. 2. We observe that the ghost stabilization (again σ = 20 here) is necessary to obtain the nice conditioning. The results with higher order Pk finite elements, k = 2, 3 are reported in Fig. 3. The
optimal convergence orders under the mesh refinement are again observed (with the order (k + 1) in the L2norm, better than in theory). The influence of the stabilization parameter σ on the accuracy of φ-FEM
with P1 and P2 finite elements is investigated at Fig. 4: the method is robust with respect to the value of
σ at least in the range [0.1, 20].
2nd test case
We now set Ω as the rectangle with corners 2π2
π2+1, π3 −π π2+1 , (0, π),− 2π2 π2+1, − π3 −π π2+1 , (0, −π) and φ(x, y) = −(y − πx − π) × (y + x/pi − π) × (y − πx + π) × (y + x/pi + π). We use φ-FEM to solve numerically Poisson-Dirichlet problem (1) in Ω with the right-hand f = 1. This test case is not consistent with Assumption 1. We want here to test φ-FEM outside of the setting where it is theoretically justified. The results with P1
and P2 finite elements are reported in Fig. 5. We observe the optimal convergence in the case k = 1 and
somewhat close to optimal convergence in the case k = 2.
3rd test case
To get further outside of the theoretically comfortable playground, we consider the problem
− div(A∇u) + u = fon Ω, u = gon Γ, (40)
with a smooth positive coefficient A, assumed to be known on O ⊃ Ω. In order to apply the φ-FEM idea to the non-homogeneous boundary conditions in (40), we assume that g is actually defined and is sufficiently smooth on O. We consider then the ansatz uh= φhwh+ gh, where gh∈ Vh(k) is an interpolant of g on Th
and wh∈ Vh(k)is the solution to the following problem generalizing (5)
Z Ωh [A∇(φhwh+ gh) · ∇(φhvh) + (φhwh+ gh)φhvh] − Z ∂Ωh A ∂ ∂n(φhwh+ gh)φhvh +σh X E∈FΓ h Z E ∂ ∂n(φhwh+ gh) ∂ ∂n(φhvh) + σh2 X T ∈TΓ h Z TL(φ hwh+ gh)L(φhvh) = Z Ωh f φhvh− σh2 X T ∈TΓ h Z Tf L(φ hvh), ∀vh∈ V (k) h (41) with L(v) = − div(A∇v) + v.
We use φ-FEM (41) to solve numerically Problem (40) on the domain Ω defined by the level-set function φ given in the polar coordinates (r, θ) by φ(r, θ) = r4(5 + 3 sin(7θ + 7π/36))/2 − 0.474, taking
O = (−1, 1)2as the surrounding domain (see [11, 12] for pictures of Ω). We choose A(x, y) = (1 + x2+ y2), adjust f so that the exact solution is given by u(x, y) = sin(x) exp(y), and take the Dirichlet data as g(x, y) = φ(x, y) exp(x) sin(y) + u(x, y) so that u = g on Γ only. The results are presented in Figure 5. We observe that the Ghost-penalty part is essential and ensures the optimal convergence of the method.
6
Conclusions and outlook
The numerical results from the last section confirm the theoretically predicted optimal convergence of φ-FEM in the H1 semi-norm. The convergence in the L2 norm turns out to be also optimal, which is
10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 100 1 1 1 2 h ku − uhk0,Ωh/kuk0,Ωh |u − uh|1,Ωh/|u|1,Ωh 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 100 101 1 1 1 2 h ku − uhk0,Ωh/kuk0,Ωh |u − uh|1,Ωh/|u|1,Ωh
Figure 1: Relative errors of φ-FEM for the 1st test case and k = 1. Left: φ-FEM with ghost penalty σ = 20; Right: φ-FEM without ghost penalty (σ = 0).
10−2 10−1 101 102 103 104 105 1 2 h Conditioning 10−2 10−1 102 103 104 105 106 107 1 4 h Conditioning
Figure 2: Condition numbers for φ-FEM in the 1st test case and k = 1. Left: φ-FEM with ghost penalty σ = 20; Right: φ-FEM without ghost penalty (σ = 0).
10−3 10−2 10−1 10−10 10−8 10−6 10−4 10−2 1 2 1 3 h ku − uhk0,Ωh/kuk0,Ωh |u − uh|1,Ωh/|u|1,Ωh 10−2 10−1 10−10 10−8 10−6 10−4 10−2 1 3 1 4 h ku − uhk0,Ωh/kuk0,Ωh |u − uh|1,Ωh/|u|1,Ωh
Figure 3: Relative errors of φ-FEM for the 1st test case. Left: k = 2; Right: k = 3.
10−510−410−310−210−1 100 101 102 103 10−2 10−1 100 101 σ h = 1.41 × 10−2 h = 7.07 × 10−3 h = 3.54 × 10−3 h = 1.77 × 10−3 10−510−410−310−210−1 100 101 102 103 10−6 10−4 10−2 100 102 104 σ h = 1.41 × 10−2 h = 7.07 × 10−3 h = 3.54 × 10−3 h = 1.77 × 10−3
Figure 4: Influence of the ghost penalty parameter σ on the relative error |u − uh|1,Ω/|u|1,Ωfor φ-FEM in
the 1st test case. Left: k = 1; Right: k = 2.
10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 100 1 1 1 2 h
kuref− uhk0,Ω/kurefk0,Ω
|uref − uh|1,Ω/|uref|1,Ω
10−2 10−1 10−8 10−7 10−6 10−5 10−4 10−3 10−2 1 2 1 3 h
kuref − uhk0,Ω/kurefk0,Ω
|uref− uh|1,Ω/|uref|1,Ω
Figure 5: Relative errors of φ-FEM for the 2nd test case. Left: k = 1; Right: k = 2. The reference solution uref is computed by a standard F EM on a sufficiently fine fitted mesh on Ω.
10−3 10−2 10−1 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 1 1 1 2 h ku − uhk0,Ωh/kuk0,Ωh |u − uh|1,Ωh/|u|1,Ωh 10−3 10−2 10−1 10−7 10−5 10−3 10−1 1 1 1 2 h ku − uhk0,Ωh/kuk0,Ωh |u − uh|1,Ωh/|u|1,Ωh
Figure 6: Relative errors of φ-FEM for the 3th test case and k = 1. Left: φ-FEM with ghost penalty σ = 20; Right: φ-FEM without ghost penalty (σ = 0).
better than the theoretical prediction. We have thus an easily implementable optimally convergent finite element method suitable for non-fitted meshes and robust with respect to the cuts of the mesh with the domain boundary. This comes at the expense of augmenting the polynomial degrees in the finite element formulation in comparison with the standard FEM and thus necessitating higher order quadrature rules. It would be interesting to investigate the effect of “under-integrating”, i.e. lowering the quadrature order, on the accuracy of the method.
Of course, the scope of the present article is very limited and academic: we only consider here the Poisson equation with homogeneous boundary conditions. An extension to non-homogeneous Dirichlet condition u = g on Γ and to a more general second order equation (40) is straightforward. It is presented (without any theoretical analysis) in the 3rd test case above. On the other hand, treating Neumann or
Robin boundary conditions is a completely different matter. We announce here an ongoing work [9], where a Neumann problem is discretized in the φ-FEM manner by introducing some auxiliary unknowns on ΩΓ
h. Future endeavors should also be devoted to more complicated governing equations and boundary
conditions.
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